ABSTRACT
In this paper, we mainly present some completely monotonic properties and new inequalities involving the k-gamma and the k-polygamma functions.
Funding statement: This work was supported by National Natural Science Foundation of China (Grant No. 11601036 and 620730453), and by the Major Project of Binzhou University (Grant No.2020ZD02).
REFERENCES
[1] Abramowitz, M.—Stegun, I. A.: Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, National Bureau of Standards, Applied Mathematics Series 55, 9th ed., Washington, 1970.Search in Google Scholar
[2] Alzer, H.: On some inequalities for the gamma and psi functions, Math. Comp. 66 (1997), 373–389.10.1090/S0025-5718-97-00807-7Search in Google Scholar
[3] Alzer, H.—Felder, G.: A Turán-type inequality for the gamma function, J. Math. Anal. Appl. 350 (2009), 276–282.10.1016/j.jmaa.2008.09.053Search in Google Scholar
[4] Alzer, H.—Jameson, G.: A harmonic mean inequality for the digamma function and related results, Rend. Semin. Mat. Univ. Padova 137 (2017), 203–209.10.4171/RSMUP/137-10Search in Google Scholar
[5] Alzer, H.: A harmonic mean inequality for the gamma function, J. Comput. Appl. Math 87(2) (1997), 195–198.10.1016/S0377-0427(96)00181-1Search in Google Scholar
[6] Alzer, H.: Inequalites for the gamma function, Proc. Edinb. Math. Soc. (2) 45(3) (2002), 589–600.10.1017/S0013091501000943Search in Google Scholar
[7] Alzer, H.: On Gautschi’s harmonic mean inequality for the gamma function, J. Comput. Appl. Math 157(1) (2003), 243–249.10.1016/S0377-0427(03)00456-4Search in Google Scholar
[8] Alzer, H.—Wells, J.: Inequality for the polygamma functions, SIAM J. Math. Anal. 29(6) (1998), 1459– 1466.10.1137/S0036141097325071Search in Google Scholar
[9] Batir, N.: On some properties of digamma and the polygamma functions, J. Math. Anal. Appl. 328(1) (2007), 452–465.10.1016/j.jmaa.2006.05.065Search in Google Scholar
[10] Berg, C.: Integral representation of some functions related to the gamma function, Mediterr. J. Math. 1(4) (2004), 433–439.10.1007/s00009-004-0022-6Search in Google Scholar
[11] Bouali, M.: A harmonic mean inequality for the q-gamma and q-digamma functions, https://arxiv.org/abs/2005.08945.Search in Google Scholar
[12] Burić, T.—Elezovć, N.: Some completely monotonic fuctions related to the psi function, Math. Inequal. Appl. 14(3) (2011), 679–691.10.7153/mia-14-57Search in Google Scholar
[13] Das, S.—Swaminathan, A.: A harmonic mean inequality for the polygamma functions, Math. Inequal. Appl. 23(1) (2020), 71–76.10.7153/mia-2020-23-06Search in Google Scholar
[14] Díaz, R.—Pariguan, E.: On hypergeometric functions and k-Pochhammer symbol, Divulg. Mat. 15(2) (2007), 179–192.Search in Google Scholar
[15] Dubourdieu, J.: Sur un théorème de M. S. Bernstein relatif à la transformation de Laplace-Stieltjes, Compos. Math. 7 (1940), 96–111.Search in Google Scholar
[16] Gautschi, W.: Some mean value inequalities for the gamma function, SIAM J. Math. Anal. 5 (1974), 282–292.10.1137/0505031Search in Google Scholar
[17] Gautschi, W.: A harmonic mean inequality for the gamma function, SIAM J. Math. Anal. 5 (1974), 278– 281.10.1137/0505030Search in Google Scholar
[18] Guo, B. N.—QI, F.: Two new proofs of the complete monotonicity of a function involving the psi function, Bull. Korean Math. Soc. 47(1) (2010), 103–111.10.4134/BKMS.2010.47.1.103Search in Google Scholar
[19] Haeringen, H. V.: Completely monotonic and related functions, J. Math. Anal. Appl. 204 (1996), 389–408.10.1006/jmaa.1996.0443Search in Google Scholar
[20] Marsden, J. E.: Basic Complex Analysis, W. H. Freeman and Company, San Fransisco, 1973.Search in Google Scholar
[21] Mehrez, M.—Sitnik, S. M.: Proofs of some conjectures on monotonicity of ratios of Kummer, Gauss and generalized hypergeometric functions, Analysis 36(4) (2016), 263–268.10.1515/anly-2015-0029Search in Google Scholar
[22] Nantomah, K.—Prempeh, E.—Twum, S. B.: On a (p,k)-analogue of the gamma function and some associated inequalities, Moroccan J. Pure Appl. Anal. 2(2) (2016), 79–90.10.7603/s40956-016-0006-0Search in Google Scholar
[23] Qi, F.—Guo, S. L.—Guo, B. N.: Complete monotonicity of some functions involving polygamma functions, J. Comput. Appl. Math. 233 (2010), 2149–2160.10.1016/j.cam.2009.09.044Search in Google Scholar
[24] Qi, F.—Sun, K. S.: A monotonicity result of a function involving the gamma functions, Anal. Math. 32(4) (2006), 279–282.10.1007/s10476-006-0012-ySearch in Google Scholar
[25] Qi, F.—Cui, R. Q.—Chen, C. P.—Guo, B. B.: Some completely monotonic functions involving polygamma functions and an application, J. Math. Anal. Appl. 310 (2005), 303–308.10.1016/j.jmaa.2005.02.016Search in Google Scholar
[26] Qi, F.—Liu, AI. Q.: Completely monotonic degrees for a difference between the logarithmic and psi functions, J. Comp. Anal. Math. 361 (2019), 366–373.10.1016/j.cam.2019.05.001Search in Google Scholar
[27] Qi, F.—Berg, C.: Complete monotonicity of a difference between the exponential and trigamma functions and properties related to a modified Bessel function, Mediterr. J. Math. 10(4) (2013), 1685–1696.10.1007/s00009-013-0272-2Search in Google Scholar
[28] Srivastava, H. M.—Choi, J. S.: Zeta and q-Zeta Functions and Associated Series and Integrals, Singapore, 2012.10.1016/B978-0-12-385218-2.00002-5Search in Google Scholar
[29] Trimble, S. Y.—Wells, J.—Wright, F. T.: Superadditive functions and a statistical application, SIAM J. Math. Anal. 20(5) (1989), 1255–1259.10.1137/0520082Search in Google Scholar
[30] Widder, D. V.: The Laplace Transform, Princeton University Press, New York, 1946.Search in Google Scholar
[31] Yang, Z. H.—Zheng, S. Z.: Monotonicity of a mean related to psi and polygamma functions with an application, J. Inequal. Appl. 2016 (2016), Art. No. 216.10.1186/s13660-016-1155-4Search in Google Scholar
[32] Yin, L.: Complete monotonicity of a function involving the (p, k)-digamma function, Int. J. Open Problems Compt. Math. 11(2) (2018), 103–108.10.12816/0049066Search in Google Scholar
[33] Yin, L.—Huang, L. G.—Song, Z. M.—Dou, X. K.: Some monotonicity properties and inequalities for the generalized digamma and polygamma functions, J. Inequal. Appl. 2018 (2018), Art. No. 249.10.1186/s13660-018-1844-2Search in Google Scholar PubMed PubMed Central
[34] Yin, L.: A monotonic properties for ratio of the generalized polygamma functions, Octogon Math. Mag. 27(1) 2019, 92–100.Search in Google Scholar
[35] Yin, L.—Huang, L.-G.—Lin, X.-L.—Wang, Y.-L.: Monotonicity, concavity, and inequalities related to the generalized digamma function, Adv. Difference Equ. 2018 (2018), Art. No. 246.10.1186/s13662-018-1695-7Search in Google Scholar
[36] Yin, L.—Huang, L.-G.—Lin, X.-L.: Complete monotonicity of some functions involving k-digamma function with application, J. Math. Inequal. 15(1) (2021), 229–238.10.7153/jmi-2021-15-17Search in Google Scholar
[37] Yin, L.: A Grübaum type inequality for k-gamma function, Turkish J. Inequal. 3(1) (2019), 28–34.Search in Google Scholar
[38] Thandapani, E.—Manuel, M. M. S.—Graef, J. R.—Spikes, P. W.: Oscillation of higher order neutral difference equation with a forcing term, Int. J. Math. Math Sci. 22 (1999), 147–154.10.1155/S0161171299221473Search in Google Scholar
[39] Yildiz, M. K.—Ocalan, O.: Oscillation results for higher order non-linear neutral delay difference equations, Appl. Math. Lett. 20 (2007), 243–247.10.1016/j.aml.2006.05.001Search in Google Scholar
© 2023 Mathematical Institute Slovak Academy of Sciences
Articles in the same Issue
- Remembering Professor Štefan Znám, 9.2.1936–17.7.1993
- Chordal and Perfect Zero-Divisor Graphs of Posets and Applications to Graphs Associated with Algebraic Structures
- Enlargements of Quantales
- Multiplicative Dependent Pairs in the Sequence of Padovan Numbers
- Dirichlet Series with Periodic Coefficients, Riemann’s Functional Equation, and Real Zeros of Dirichlet L-Functions
- On the Rational Parametric Solution of Diagonal Quartic Varieties
- Theory of Certain Non-Univalent Analytic Functions
- Initial Coefficients and Fekete-Szegő Inequalities for Functions Related to van der Pol Numbers (VPN)
- A Conjecture on H3(1) for Certain Starlike Functions
- Coefficient Problems of Quasi-Convex Mappings of Type B on the Unit Ball in Complex Banach Spaces
- Complete Monotonicity and Inequalities Involving the k-Gamma and k-Polygamma Functions
- Study of Oscillation Criteria of Odd-Order Differential Equations with Mixed Neutral Terms
- On a System of Difference Equations Defined by the Product of Separable Homogeneous Functions
- On the Existence of Bi-Lipschitz Equivalent Metrics in Semimetric Spaces
- The Lehmann Type II Teissier Distribution
- Asymptotic Predictive Inference of Negative Lower Tail Index Distributions
- On Numerical Problems in Computing Life Annuities Based on the Makeham–Beard Law
- The Rational Zero-Divisor Cup-Length of Oriented Partial Flag Manifolds
Articles in the same Issue
- Remembering Professor Štefan Znám, 9.2.1936–17.7.1993
- Chordal and Perfect Zero-Divisor Graphs of Posets and Applications to Graphs Associated with Algebraic Structures
- Enlargements of Quantales
- Multiplicative Dependent Pairs in the Sequence of Padovan Numbers
- Dirichlet Series with Periodic Coefficients, Riemann’s Functional Equation, and Real Zeros of Dirichlet L-Functions
- On the Rational Parametric Solution of Diagonal Quartic Varieties
- Theory of Certain Non-Univalent Analytic Functions
- Initial Coefficients and Fekete-Szegő Inequalities for Functions Related to van der Pol Numbers (VPN)
- A Conjecture on H3(1) for Certain Starlike Functions
- Coefficient Problems of Quasi-Convex Mappings of Type B on the Unit Ball in Complex Banach Spaces
- Complete Monotonicity and Inequalities Involving the k-Gamma and k-Polygamma Functions
- Study of Oscillation Criteria of Odd-Order Differential Equations with Mixed Neutral Terms
- On a System of Difference Equations Defined by the Product of Separable Homogeneous Functions
- On the Existence of Bi-Lipschitz Equivalent Metrics in Semimetric Spaces
- The Lehmann Type II Teissier Distribution
- Asymptotic Predictive Inference of Negative Lower Tail Index Distributions
- On Numerical Problems in Computing Life Annuities Based on the Makeham–Beard Law
- The Rational Zero-Divisor Cup-Length of Oriented Partial Flag Manifolds